Let be a real vector space and be a real function on satisfying the following three conditions:

The function is called a norm on and the pair is called a normed space [1]. A norm can be viewed as a generalization of the concept of length within a vector space.

A set of p-summable sequences, with , is one of vector spaces in functional analysis. Here, the vector space is denoted by which contains all sequences of real numbers for which (or while ). Usually, is equipped with the usual norm

so that becomes a normed space.

We also introduce an n-norm. Consider a vector space with . By [2], we give a mapping such that it has four conditions:

We call that is an n-norm and the pair is an n-normed space. On , for simplification, we occasionally express the n-norm as

for every .

In functional analysis, the concept of n-normed spaces generalizes normed spaces. This idea was studied by various researchers who examined different properties of n-normed spaces. The notion of n-normed spaces was first introduced by Misiak [3], and subsequent contributions were made by Dutta [4], Gunawan and Mashadi [5], Gunawan [6], Kim and Cho [7], Lewandowska [8], and many others. In the specific case where n = 2, the notion of 2-normed spaces was introduced by Gähler in the mid-1960s (see [9]). This concept was further developed by Diminnie, Gähler, White [10, 11], among others.

We can add an extra property to the n-norm, as described in the following proposition.

Proposition 1.1.On , we have a property

for every and for every scalar .

The proof of the above proposition is quite straightforward, relying only on the triangle inequality, the homogeneity property, and the concept of linearly dependent vectors. We leave the verification to the reader.

For where , we recall n-norm which is defined by Gunawan [2]. That is a mapping which is stated by the following equation

for every . For , that is

for every . Note that means an

absolute value of . In this article, the n-norm defined by Gunawan (4) is referred to as the usual n-norm.

We now describe that a space , equipped with a norm, is called a complete space (or Banach space) if every Cauchy sequence in converges to a limit within the space. In the context of an n-normed space , a sequence converges to if and only if

for all as . Likewise, the sequence is Cauchy if and only if

for all as . An n-Banach space is an n-normed space where every Cauchy sequence converges. It was shown in [1] and [2] that both the usual norm and the usual n-norm result in complete spaces, meaning that both are a Banach spaces and an n-Banach spaces respectively. On , Konca et al. [12] conducted deeply investigation into the equivalence of two definitions of convergence (two definitions of Cauchy sequences).

In 2000, Gunawan introduced the norm induced by the usual n-norm and the first n standard basis vectors in [2]. Later, in 2013, he and his students refined this definition by using any set of n linearly independent vectors in [13]. This prompts the question: can we define an n-norm that is induced by a norm? For , defining a 2-norm on is fairly straightforward, along with the corresponding 2-inner product. For with , Konca et al. introduced a 2-inner product and derived a 2-norm (see [14]). They also investigated the completeness of these spaces. In 2016, Konca and Idris defined a 2-norm using the usual norm on along with a set of bounded linear functionals [15], and later simplified this definition in [16]. They also explored the relationships between Gähler's 2-norm [9] and Gunawan's 2-norm (the usual 2-norm) [2]. In this work, we extend the approach of Konca and Idris to the case of 2 \end{document} ]]>. Additionally, we examine the relationship between their definition and the two different definitions of n-norms provided by Gähler and Gunawan. Furthermore, we investigate the significant role of bounded linear functionals on the space where . Our findings confirm that all the fundamental properties of the n-norm on , constructed in this study are consistent with those discussed in the article "New 2-norms formed on by bounded linear functionals".

For further details on bounded linear functionals, we refer to [16]. Let be a normed space and a bounded linear functional. The set of all bounded linear functionals on , known as the dual space of , is denoted by . Now, observe that for every and , we have the inequality , or equivalently, . Consequently, we obtain the following result:

On , the usual norm is defined as in (1). Let with 0 \end{document} ]]> and . For every , we define the functional as follows:

This functional is linear, and by applying Hölder's inequality, for every , we obtain

Thus, we conclude that is a bounded linear functional. We then collect all such bounded linear functionals into the set

Here, represents the dual space of , consisting of all bounded linear functionals on . As noted in [1], we have a norm , and we say that is a normed space.

Based on the previous description, we now present a definition of another norm on using sets of bounded linear functionals. Specifically, we define

for every . Readers can verify that this definition indeed defines a norm. Next, we explore its relationship with the usual norm. For , let be defined as , where sign . We have

Additionally, the norm of the functional is computed as

Thus, we obtain

Since the inequality in equation (6) holds, we conclude that

for every . This shows that the norms and are equivalent.

We begin by recalling Gähler's definition of the n-norm on , where . Let be the dual space consisting of all bounded linear functionals, where . According to Gähler, the n-norm is defined by

for every .

Lemma 3.1. For and where , we have

Proof. Take arbitrary and where .Let

Using the properties of determinant and applying Laplace expansion, we get

Check that

So, we obtain

Moving to , we obtain

Next, moving to where goes to

We repeat the same process for to get

Applying the homogeneity property of summability,

holds. The proof is thereby completed.

By the above lemma, we have

and we also use it for (8)

In (8), there is a slight modification to Gähler's definition of the n-norm. Here, we add the condition for to obtain the supremum . This condition is required since defining a new n-norm involves a division by . Now, take arbitrary and let

.

Using (4), we get

We proceed to define a mapping based on the usual -norm on and the set of bounded linear functionals . The definition is given as follows.

for every .

Since is an -norm, then we can use its properties to verify that (10) is an n-norm. Readers may check its properties. Now, we represent (9) in a form

where

Using Laplace expansion, (11) becomes

Consequently, (10) becomes

From the above discussions and previous studies, it is clear that inducing a norm from the n-norm can be done effectively. Now, we also need to attempt to induce an n-norm from a norm on . Konca and Idris have defined the 2-norm from the norm (see [15, 16]). Furthermore, the above definition shows the induction of the n-norm from the -norm. In the last section, we can define the n-norm recursively based on the norm.

Taking arbitrary , we define 2-norm

Next, we define 3-norm

Using Laplace expansion, verify that

Now, we have

Repeat by defining the 4-norm up to the -norm and the last, we get the definition of the n-norm

Interestingly, we also examine the relationship between and other n-norms mentioned above.

Proposition 4.1. Let . We have an inequality

for every with and .

Proof. Recall (8). For arbitrary , we have

where . By Hölder's inequality, we obtain

Observe also that, by the triangle inequality and Hölder's inequality,

holds, where . As a result,

Moreover, it can be observed that

So, we have

Accordingly, the four definitions of n-norms give rise to an inequality

The proof is complete.

With the results from Theorem 3.3 and Proposition 4.1, on , we get

Using fixed linear independent vectors , we also define

for every . Since (16) holds, .

Corollary 4.2. The spaces and are an n-Banach space and a Banach space respectively.

Proof. Use the method similar to the proof of Corollary 3.4 and the proof of Corollary 3.6.

We have defined several n-norms and norms on and demonstrated their interrelationships. This investigation could similarly be extended to , or more generally to as a normed space with an appropriate set of bounded linear functionals. In such cases, new challenges may arise, potentially leading to different results.

The author is deeply thankful to the anonymous reviewer(s) for their constructive comments and recommendations, which greatly aided in refining this article.